11-24 할리데이 11판 솔루션 일반물리학

$$\put \begin{cases} A : \text{Disk}\\ B : \text{Man}\\ \end{cases}$$ $$\begin{cases} R&=1.20\ut{m}\\ m_A&=200\ut{kg}\\ k_A&=91.0\ut{cm}=0.910\ut{m}\\ m_B&=44.0\ut{kg}\\ v_{Ai}&=0\\ v_{Bi}&=3.00\ut{m/s}\\ \end{cases}$$ $$\ab{a}$$ $$k_A=\sqrt\frac{I_A}{m_A},$$ $$\begin{aligned} I_A&=m_A{k_A}^2\\ &=\frac{8281}{50}\ut{kg\cdot m^2}\\ &=165.62\ut{kg\cdot m^2}\\ &\approx 166\ut{kg\cdot m^2}\\ \end{aligned}$$ $$\ab{b}$$ $$\begin{aligned} \vec L&=\vec r\times \vec p\\ &=m(\vec r\times \vec v)\\ \end{aligned}$$ $$\begin{aligned} L_{Bi}&=m_BRv_{Bi}\\ &=\frac{792}{5}\ut{kg\cdot m^2/s}\\ &=158.4\ut{kg\cdot m^2/s}\\ &\approx 158\ut{kg\cdot m^2/s}\\ \end{aligned}$$ $$\ab{c}$$ $$\begin{aligned} \Sigma I&= I_A+I_B\\ &=m_A{k_A}^2+m_BR^2\\ &=\frac{11449}{50}\ut{kg\cdot m^2}\\ &=228.98\ut{kg\cdot m^2}\\ \end{aligned}$$ $$\Delta \Sigma \vec L=0,$$ $$\begin{aligned} 0 &=\Delta L_A+\Delta L_B\\ &= L_{Af}+ (L_{Bf}-L_{Bi})\\ &= L_{f}-L_{Bi}\\ &=\omega_f\Sigma I -L_{Bi}\\ \end{aligned}$$ $$\begin{aligned} \omega_f&=\frac{L_{Bi}}{\Sigma I}\\ &=\frac{7920}{11449}\ut{rad/s}\\ &\approx 0.6917634727923836\ut{rad/s}\\ &\approx 0.692\ut{rad/s}\\ \end{aligned}$$